ALBERT

Description: More than 30 years ago, Diffusion-Limited Aggregation (DLA) has been studied as mechanism to generate structures sharing similarities with real-world cities. Recently, a stochastic gravitation model (SGM) has been proposed for the same purpose but representing a completely different mechanism. Both approaches have advantages and disadvantages, while e.g. the dendrites emerging via DLA are visually very different from real-world cities, the SGM does not preserve undeveloped locations close to the city center. Here we propose a unification of both mechanisms, i.e. a particle moves randomly and turns into an urban site with a probability that depends on the proximity to already developed sites. We study the cluster size distributions of the structures generated by both models and find that SGM generates more balanced distributions. We also propose a way to assess to which extent the largest cluster is a primate city and find that in both models, beyond certain parameter value, the size of the largest cluster becomes inconsistent with being drawn from the same distribution of remaining clusters.

Description: Many physical, biological, and social systems exhibit emergent properties arising from their components’ interactions (cells). In this study, we systematically treat every-pair interactions (a) that exhibit power-law dependence on the Euclidean distance and (b) act in structures that can be characterized using fractal geometry. It can represent the two-body interaction potential, the heat flux between two parts of a structure, friendship strength between two people, etc.. We analytically derive the average intensity of influence that one cell has on the others or, conversely, receives from them. This quantity is referred to as the mean interaction field of the cells, and we find that (i) in a long-range interaction regime, the mean interaction field increases following a power-law with the size of the system, (ii) in a short-range interaction regime, the field saturates, and (iii) in the intermediate range it follows a logarithmic behavior. To validate our analytical solution, we perform numerical simulations. For long-range interactions, the theoretical calculations align closely with the numerical results. However, for short-range interactions, we observe that discreteness significantly impacts the continuum approximation used in the derivation, leading to incorrect asymptotic behavior in this regime. To address this issue, we propose an expansion that substantially improves the accuracy of the analytical expression. We discuss applications of the every-pair interactions system proposed, and one of them is to explore a framework for estimating the fractal dimension of unknown structures. This approach offers an alternative to established methods such as box-counting or sandbox methods. Overall, we believe that our analytical work will have broad applicability in systems where every-pair interactions play a role.